Sometimes, there are deadlines for events and sometimes they are extended. In case of such an event, is it correct to say "extension" of last date? But, how can a date itself be extended? I guess relaxed is a more appropriate word.

I don't know why the downvoter voted, but this sounds way too local to me. A date can't be literally extended (it is a point in time) but metonymically it can (the span of time to reach that point). So 'extended' is fine and better than 'relax'. Also 'relaxation of the due date' works, but 'relaxed the due date' (verb without preposition here.) Stick with 'extended' though.
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MitchFeb 15 '13 at 15:22

4 Answers
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With deadlines, relaxed can be used, but extended is both reasonable and more common. Both mean the deadline was changed to a different date (the date itself was unaffected, and e.g. the 27th of February would remain the 27th of February).

With events like a show for which there were say showings from the 13th to the 20th that was then changed to run from the 13th to the 24th, we would not normally use relaxed (unlike a deadline, it doesn't ease anyone's workload to have this happen), but would use extended.

We might say the last date was extended, again this doesn't affect the date itself, but it affects the event; The event has a property that is it's last date, and that property was extended. This is someone would mean if they said "the last date was extended for another 3 days".

I think you've hit on the most important factor - the fact that the term date isn't irrevocably tied in to a fixed position on the calendar: date 1 (dt) n. 1. a. Time stated in terms of the day, month, and year. b. A statement of calendar time, as on a document. [the above unalterable unless we rejig the calendar itself] 3. a. A particular point or period of time at which something happened or existed, or is [']expected['] to happen. [and the latter subject to change, especially when that would be least convenient] (AHDEL in the main)
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Edwin AshworthFeb 15 '13 at 15:49

To extend something is to stretch it. In this case, it is the time that is allowed for some kind of activity that must take place prior to the event.

The committee has extended the submission period for manuscripts by another week.

Ticket sales have been extended through the week of July 1.

But extend can also mean to push something outward— for example, changing a date to a later date.

They extended the registration deadline to November.

The librarian extended the due date to the 12th.

The cutoff for applications was extended to close-of-business on Tuesday.

In this sense, I wouldn't say it's strictly wrong to speak of a date by itself being extended, but I don't think this usage is typical among native speakers.

The phrase dates extended means that certain events that took place on certain days will now also take place on additional days (for example, a performer having additional shows), and should not be confused.

Due dates are not relaxed. To relax something is to make it less rigid or strict, so one can speak of the rule or requirement that something be completed by a certain date being relaxed, but not the date itself.

The professor relaxed her term paper guidelines; they can now be turned in after the public holiday or by e-mail.